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EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Published online by Cambridge University Press:  07 March 2019

JAROSLAV NEŠETŘIL  and
PATRICE OSSONA DE MENDEZ
JAROSLAV NEŠETŘIL
Affiliation:
COMPUTER SCIENCE INSTITUTE OF CHARLES UNIVERSITY (IUUK AND ITI) MALOSTRANSKÉ NÁM.25, 11800PRAHA 1, CZECH REPUBLIC E-mail: nesetril@iuuk.mff.cuni.cz
PATRICE OSSONA DE MENDEZ
Affiliation:
CENTRE D’ANALYSE ET DE MATHÉMATIQUES SOCIALES (CNRS, UMR 8557) 190-198 AVENUE DE FRANCE, 75013 PARIS, FRANCE and COMPUTER SCIENCE INSTITUTE OF CHARLES UNIVERSITY (IUUK) MALOSTRANSKÉ NÁM.25, 11800PRAHA 1, CZECH REPUBLIC E-mail: pom@ehess.fr
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Abstract

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:

  1. 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.

  2. 2. A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.

In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Keywords

Primary 03C3003C9805C99Secondary 03C4528E1503C50first-order propertieslimit statisticsconvergence of graphsstructural limittotally Borel graphmodeling
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 452 - 472
Copyright
Copyright © The Association for Symbolic Logic 2019 

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